Accardi L., Freudenberg W., Obya M. (Eds.) Quantum Bio-informatics IV: From Quantum Information to Bio-informatics

190

be

a

dynamical

semigroup, where

B*(Ji)

denotes

th

e real vector space of all

self-adjoint

operators

on

Ji

.

If

one introduces

the

scalar

product

of opera-

tors

A, B by

the

formula (A,

B)

=

Tr(A*

B),

then

B*(Ji)

can

be

considered

as yet

another

inner

product

space,

namely

the

so-called Hilbert-Schmidt

space

with

the

norm

defined by

II

p

112=

Tr(p*p).

Stat

es

of

the

system

are

described by density

operators

p E S(Ji), where

S(Ji)

:=

{p E B*(Ji); p

2:

0,

Tr p =

I}.

(8)

Usually one assumes

that

the

family

of

linear

superoperators

(t)

satisfies

(1)

(t)

is

trace

preserving, t E

~~,

(2)

II

(t)p

II

::;

II

p

II

for all p E

B*(Ji)

,

(3)

(h)

0

(t2)

=

(h

+ t2),

for all t1,

t2

in

~~,

and

ift

-+

0,

then

lim

(t)

=

If.

Since such defined

(t)

is a contraction,

it

follows from

the

Hille-Yosida

theorem

that

there

exists

a linear

superoperator

lK

:

B*(Ji)

-+

B*(Ji)

such

that

(t)

= exp(tlK) for

all t

2:

0

and

dp(t)

=1V

()

dt

!R.p

t ,

(9)

where p(t) =

(t)p(O).

One

should stress

that

the

above conditions for

semigroup

( t) imply preservation

of

positivity

of

density

operators

,

p(O)

2:

o

=?

p(t) =

(t)p(O)

2:

0 for all t E

~~.

Now,

the

equation

(9) (usually

called

the

master

equation) defines

an

assignment

(th

e

trajectory

of

p(O))

~~

:3

t

t--->

p(t) E S(Ji),

(10)

provided

that

we know

the

initial

state

of

the

system

p(O)

E S(Ji).

The

fundamental

question

of

the

stroboscopic

tomography

reads:

What

can

we

say

about

the

trajectories

(initial

state

p(O))

if

the

only information

about

the

system

in question is given by

the

mean

values

(11)

of, say,

r lin

ea

rly

independent

self-adjoint

operators

Q1,

...

,

Qr

at

some

instants

t1,

...

, t

p

,

where r < d

2

- 1

and

tj E

[0,

TJ

for j =

1,

...

,p,

T>

O.

In

other

words,

the

problem

of

the

stroboscopic

tomography

consists

in

the

reconstruction

of

the

initial

state

p(O),

or

a

current

state

p(t), for a

ny

t E

~~,

from known

expectation

values (11). To

be

more precise we introduce

the

following description. Suppose

that

we

can

prepare

a

quantum

system

repeatedly

in

the

same

initial

state

and

we make a series

of

experiments

such

that

we know

the

expectation

values EQ(tj) = Tr(Qp(tj)) for a fixed

191

set

of

observables Q1,'"

,Qr

at

different

time

instants

h <

t2

< ... < tp'

The

basic

question

is:

Can

we find

the

expectation

value

of

any

other

operator

Q E B*(7t),

that

is

any

other

observable from B*(7t) , knowing

the

set

of

measured

outcomes

of

a given

set

Q1,""

Qr

at

t1,

...

, t

p

, i.e.

knowing

Ej(tk)

for j = 1,

...

, r

and

where 0 S

t1

<

t2

< ... <

tp

S

T,

for

an

interval

[0,

T]?

If

the

problem

under

consideration

is

static,

then

the

state

of

ad-level

open

quantum

system

(a

qudit)

can

be

uniquely

determined

only if r =

d

2

-

1

expectation

values

of

linearly

independent

observables

are

at

our

disposal. However, if we assume

that

we know

the

dynamics

of

our

system

i.e.

we

know

the

generator

lK

or

lL

:=

(lK)*

(in

the

Heisenberg

picture)

of

the

time

evolution,

then

we

can

use

the

stroboscopic

approach

based

on

a discrete

set

of

times

t

1

, ... , tp' In general, we use

the

term

"state-

tomography"

to

denote

any

kind

of

state-reconstruction

method.

With

reference

to

the

terminology used

in

system

theory, we

introduce

the

following definition: A d-level

open

quantum

system

§ is

said

to

be

(Q1,

...

,Qr

)-reconstructible

on

the

interval

[0,

T],

if for every two

trajec-

tories defined

by

the

equation

(9)

there

exists

at

least

one

instant

£ E

[0,

T]

and

at

least

one

operator

Qk E

{Q1,""

Qr}

such

that

(12)

The

above definition is equivalent

to

the

following

statement.

A d-level

open

quantum

system

§ is (Q1,

...

, Qr )-reconstructible

on

the

interval

[0,

T], iff

there

exists

at

least

one

set

of

time

instants

0 <

t1

< ... <

tp

S T such

that

the

state

trajectory

can

be

uniquely

determined

by

the

correspondence

(13)

for i = 1,

...

,

rand

j = 1,

...

,p.

Let

us observe

that

in

the

above definition

of

reconstructibility

we dis-

cuss

the

problem

of

verifying

whether

the

accessible

information

about

the

system

in

question is sufficient

to

determine

the

state

uniquely

and

we

do

not

insist

on

determining

it

explicitly.

The

positive

dynamical

semi

group

{(t),

t E

~~}

is

determined

by

the

generator

lK

: B*(7t)

--->

B*(7t)

(the

Schrodinger

picture)

and

it

is

the

generator

lL

of

the

semigroup

in

the

Heisenberg

picture

by

the

duality

relation

Tr[Q(lKp)] = Tr[(lLQ)p].

(14)

192

For a given set

of

observables

Ql,""

Qr,

the

subspace

spanned

on

the

operators

Qi,

lLQi,

...

,

(lL)k-lQi,

will

be

denoted

by

(15)

as

the

Krylov

subspace

in

the

Hilbert-Schmidt

space

B*(H).

If

k =

JL,

where

JL

is

the

degree

of

the

minimal

polynomial

of

the

generator

lL,

then

the

subspace

JCI-'(lL,

Qi)

is

an

invariant

subspace

of

the

superoperator

lL

with

respect

to

Qi'

It

can

be

easily seen

that

the

subspace

JCI-'(lL,

Qi)

is

essentially

spanned

on

all

operators

of

the

form (lL)kQi' where k = 0,

1,

....

Furthermore,

it

is

the

smallest invariant

subspace

of

the

superoperator

lL

containing

Qi

(i.e.

the

common

part

of

all

invariant

subspaces of

the

oper-

ator

lL

containing

Qi)'

One

can

now formulate

the

sufficient conditions for

the

reconstructibility

of

a d-level

open

quantum

system

(c.f.

6,7).

Let

§

be

a d-level

open

quantum

system

with

the

evolution governed

by

an

equation

of

the

form Q(t) = lLQ(t)

(the

Heisenberg

picture),

where

lL

is

the

generator

of

the

dynamical

semigroup \f!(t) = exp(tlL). Suppose

that,

by

performing

measurements,

the

correspondence

(16)

can

be

established

for fixed observables

Ql,

...

,Qr

at

selected

time

instants

t

l

,

...

, tp.

The

system

§ is

(Ql,""

Qr)-reconstructible

if

(17)

The

above

condition

has

been

obtained

by

using

the

polynomial repre-

sentation

of

the

semigroup \f!(t). Indeed,

if

JL(A,

lL)

denotes

the

minimal

polynomial

of

the

generator

lL

and

JL

= degJL(A,lL),

then

\f!(t) = exp(tlL)

can

be

represented

in

the

form

1-'-1

\f!(t) = L Qk(t)lL

k

,

(18)

k=O

where

the

functions Qk(t) for k = 0,

...

,

JL

- 1

are

particular

solutions

of

the

scalar linear differential

equation

with

characteristic polynomial

JL(A,

lL).

Since

the

functions Qk(t)

are

mutually

independent,

therefore for

arbitrary

T > °

there

exists

at

least

one

set

of

moments

t

l

,

...

,tl-'

(JL

= deg

JL(A,

lL))

such

that

(19)

193

and

det[ak(tj)]

-=I

O.

Taking into account

these

conditions one finds

that

the

state

p(O)

can

be

determined

uniquely if

operators

of

the

form

(20)

for l =

1,

...

,

rand

k =

0,1,

...

span

the

space B*(1t).

In

other

words,

we

can

say

that

p(O)

can

be

determined

if vectors (20)

constitute

a

frame

in

Hilbert-Schmidt space B*(1t) or, equivalently, if Krylov subspaces

K!l(lL,

Qz)

for l =

1,

...

,r

constitute

a fusion frame

in

B*(1t)

15.

The

question

of

an

obvious physical interest is

to

find

the

minimal num-

ber

of

observables

Ql,""

Qry

for which a d-level

quantum

system

§

with

a

fixed

generator

lL

can

be

(Ql ,

""

Qry)-reconstructible.

It

can

be

shown

that

for

an

d-level

generator

there

always exists a

set

of observables

Ql,""

Qry,

where

TJ:=

max

{dim

Ker().IT

-lLn,

AEa(lL)

(21)

such

that

the

system

is

(Ql,

...

,Qry)-reconstructible

9.

Moreover, if

we

have

another

set

of observables

Ql,

...

,Qry

such

that

the

system

is

(Ql,

...

,Qry)-

reconstructible,

then

r;

;:::

TJ·

The

number

TJ

defined by (21) is called

the

index

of

cyclicity

of

the

quantum

open

system

§

9.

The

symbol (j(lL)

in

(21) denotes

the

spectrum

of

the

superoperator

lL.

4.

Algebraic

approach

to

identification

problems

In

this

section

we

will discuss some problems

of

estimation

of

quantum

states

when

the

Krylov subspaces playing such

an

important

role

in

the

stroboscopic

tomography

are

replaced by some

sub

algebras

of

the

Hilbert-

Schmidt space

B*(1t).

Just

as

the

fundamental

theorem

of algebra ensures

that

every linear

operator

acting

on

a finite dimensional complex

Hilbert

space

has

a nontrivial invariant subspace,

the

fundamental theorem

of

non-

commutative algebra

asserts

the

existence

of

invariant subspaces

of

1t for

some families

of

operators

from B*(1t).

It

is

an

obvious observation

that

an

algebra

generated

by

any

fixed

operator

Q

and

the

identity

on 1t

can

not

be

equal

to

B*(1t).

This

statement

is

based

on

the

Hamilton-Cayley

theorem. However,

already

for two

operators

Ql,

Q2

and

the

identity

we

can

have

Alg(I,

Ql,

Q2)

=B(1t)

(for details cf. below

and

the

In

general,

the

famous

Burnside's

theorem

states

that

an

operator

alge-

bra

on

a finite-dimensional vector space

with

no nontrivial subspaces

must

be

the

algebra

of

all linear

operators.

In

the

sequel we will use

the

following

version

of

this

theorem:

194

Fundamental theorem

of

noncommutative algebras.

If

A is a

proper

subalgebra

of

B*(H)

containing

identity,

and

the

dimension

of

the

Hilbert

space

H is

greater

or

equal

to

2,

then

A

has

a

proper

nonzero

invariant

subspace

in H (i.e.,

the

subspace

is

invariant

for all

members

Q

of

A).

We will

apply

the

above

theorem

for

the

following problem. Given a

set

F = {Q1, ... ,

Qr}

of

observables, we would like

to

establish

conditions,

when

the

operators

Q1, ... ,

Qr

generate

the

whole

algebra

B(H)

.

In

other

words,

we

want

to

determine

whether

every

element

in

B(H)

can

be

represented

in

the

form

7f(Q1,

...

,

Qr),

where

7f

is a

noncommutative

polynomial.

Let

us observe

that

according

to

the

fundamental theorem if A is a

sub

algebra

of

the

full complex

algebra

B(H),

then

a

nontrivial

invariant

subspace

in

H exists if

and

only

if

dimA

<

dimB(H).

(22)

If

a

set

of

generators

of A is known,

then

the

above

inequality

can

be

verified

by

a finite

number

of

arithmetic

operations.

The

procedures

possessing

such

property

are

called effective. A very

important

example

of

an

effective

procedure

can

be

formulated

when

we discuss

the

problem

of

the

existence a

common

one-dimensional

invariant

subspace

for a

pair

of

operators

Q1, Q2.

In

other

words, we ask

about

a

common

eigenvector for

two

operators

Q1,

Q2.

An

answer

to

this

question

is given

by

the

following

procedure.

Let

the

symbol

[Q1,

Q2J

denote,

as usual,

the

commutator

of

the

operators

Q1,

Q2.

Then

a

common

eigenvector for

Q1

and

Q2

exists, if

and

only if,

the

subspace

K

of

H defined

by

d-1

K

:=

n Ker[Q{,

Q~J

j=l

k=l

(23)

where

d =

dim

H, satisfies

the

condition

dim

K > 0

(this

is

the

so-called

Shemesh

criterion).

A

short

proof

of

this

condition

is possible.

First

of

all,

let

us observe

that

if

J'Ij!)

is a

common

eigenvector

of

the

operators

Q1

and

Q2, i.e.,

(24)

then

J'Ij!)

belongs

to

Ker[Q{,

Q~J

for all

j,

k

greater

then

1.

This

fact

and

the

inequality

dim

K > 0

means

that

the

gist

of

the

She

mesh

condition

is

in

observation

that

the

subspace

K is

invariant

under

Q1

and

Q2.

Indeed,

if

J'Ij!)

belongs

to

K,

then

by

the

definition

of

subs

paces Ker[Q{,

Q~J

one

can

check

that

Q1J'Ij!)

E K

and

Q2J'Ij!)

E

K.

Now,

let

us choose a basis for K

and

195

extend

it

to

a

basis

in

H.

We

then

observe

that

there

exists

a

nonsingular

matrix

S

such

that

matrices

SQlS-l

and

SQ2S-l

have

block-triangular

forms

and

the

submatrices

which

correspond

to

subspace

JC

commute.

This

means

that

these

submatrices

have

a

common

eigenvector

and

therefore

the

same

is

true

for

Ql

and

Q2.

D.

Shemesh

16

has

observed

that

the

condition

dim

JC

> 0 is

equivalent

to

the

singularity

of

the

matrix

d-l

M:=

2)Q{,Q~l*[Q{,Q~]'

j=l

k=l

(25)

where

*

denotes

complex

conjugate

transpose.

For

our

purposes,

on

the

basis

of

Burnside's

theorem,

more

interesting

is

the

case

when

matrices

Ql,

Q2

do

not

have

common

eigenvectors

and

the

algebra

A(Ql,

Q2)

gen-

erated

by

them

coincides

with

B(H).

This

situation

may

be

expressed

by

the

following

inequality

detM>O,

(26)

which

can

be

checked

by

an

effective

procedure,

that

is,

by

a finite

number

of

arithmetic

operations.

It

is obvious,

that

the

matrix

M is

in

general

semipositive

definite,

and

the

above

condition

means

the

strict

positivity

ofM.

5.

Examples.

Low

dimensional

cases

Now,

in

order

to

illustrate

algebraic

methods

in

estimation

problems,

we

will discuss

some

algebraic

procedures

in

low

dimensional

cases.

For

quan-

tum

systems

of

qubits

and

qutrits

one

can

formulate

an

explicit form

of

some conditions

in

a

matrix

form which is

sometimes

more

transparent

then

the

general

operator

form.

We

will use

the

so-called vee operator

pro-

cedure

which

transforms

a

matrix

into

a

vector

by

stacking

its

columns

one

underneath

the

other.

It

is well known,

that

the

tensor

product

of

matrices

and

the

vec

operator

are

intimately

connected.

If

A

denotes

a d x d

matrix

and

aj

its

j-th

column,

then

vec A is

the

d

2

-dimensional

vector

constructed

from

aI,

... , ad. Moreover if A,

B,

C

are

three

matrices

such

that

the

matrix

product

ABC

is well defined,

then

vec(ABC)

= (C

T

® A) vec

B.

(27)

In

the

above

formula

C

T

denotes

the

transposition

of

the

matrix

C.

In

particular

we

have

196

vec A =

(IT

0

A)

vec

IT

= (AT o

IT)

vec

IT.

(28)

Let

us agree

that

when we

say

that

a

set

of

matrices

generates

the

set

B(H),

we

are

thinking

about

B(H)

as

an

algebra, while when we say

that

a

set

of

matrices

forms a basis for

B(H)

,

we

are talking

about

B(H)

as a

ve

ctor

space.

For qubits,

that

is for two-dimensional Hilbert space, one can show by

a

direct

computation

that

det(vec

IT,

vec

Q1,

vec

Q2,

vec(Q1Q2)) = det([Q1,

Q2]),

and

det(veclI, vec

Q1

, vec

Q2

, vec[Q1 '

Q2])

= 2 det([Q1,

Q2]),

(29)

where

on

the

left

hand

side we have

the

determinants

of

the

4 x 4

matrices

and

on

the

right

hand

sides

[Q1,

Q2]

denotes

the

commutator

of

the

two

2 x 2 matrices.

From

the

last

equality

it

follows,

that

if

matrices

IT

,

Q1,

Q2

and

[Q1

,

Q2]

are

linearly

ind

e

pendent,

then

the

algebra

which is

spanned

by

them

has

the

dimension 4, so

Q1,

Q2

and

IT

generate

B(H).

In

other

words, two

operators

Q1

,Q2

and

the

identity

generate

B(H)

if

and

only if

the

matrix

[Q1

,

Q2]

has

th

e

determinant

different from zero.

In

a similar way one

can

show

that

the

matrices

QI,

Q2, Q3,

such

that

no two

of

them

generate

B(H),

can

generate

B(H)

if

and

only if

the

double

commutator

[Q1,

[Q2,

Q3ll is

invertible.

In

general,

the

matrices

Q1,

...

,

Qr

generate

B(H)

iff

at

least

one of

the

commutators

[Qi,

Qj]

or

double

commutators

[Qi,

[Qj,

Qkll

is

invertible

17.

In

the

case of

qutrits,

that

is for a three-dimensional Hilbert space,

one

can

show by direct calculation

that

if

[Q1,

Q2]

is invertible

and

W([Q1

,

Q2])

=1=

0, where for Q E

B(H)

the

symbol

w(Q)

denotes

the

lin-

ear

term

in

the

characteristic polynomial

of

Q,

then

one

can

construct

an

explicit basis for

B(H).

Indeed, if

Q1,

Q2

belong

to

B(H),

and

(dim

H)

= 3,

then

the

determinant

of

the

9-dimensional

matrix

n build from vec transfor-

mations

oflI,

Q1, Q2,

Qi

,

Q~,

Q1

Q2,

Q2Q1,

[Q1, [Q1,

Q2

ll

,

[Q2, [Q2,

Q1ll satis-

fies

the

equality

(30)

That

is, if det([Q1,

Q2])

=1=

0

and

w(Q)

=1=

0,

then

the

columns

of

the

matrix

n correspond

to

a basis for

B(H).

Of

course, one

can

also use

the

Shemesh criterion

to

charact

erize pairs

of generators for

B(H),

where

dim

H = 3.

197

References

1.

W.

Thirring,

Quantum Mathematical Physics, (Springer, 2001).

2.

W.

Band,

J.

L.

Park

,

Am.

J.

Phys. 47, 188 (1979).

3. M. A. Nielsen,

1.

Chuang,

Quantum Computation and Quantum Information,

(Cambridge

Univ.

Pr

ess., 2000).

4.

K.

Kr

aus,

Ann.

Phys.

64

, 119 (1971).

5.

S.

Weigert, in New Insight

in

Quantum Mechanics

Eds

. H.-D.

Doebner

et

al.

(Singapor

e:

World Scientific, 2000).

6.

A. Jamiolkowski, Rep. Math. Phys. 5, 415 (1975).

7.

A. Jamiolkowski, Internat.

J.

Theoret. Phys.

22,

369 (1983).

8.

R.

T.

Farouki

et

al., Geometriae Dedicata 85, 283 (2001).

9.

A. Jamiolkowski,

in

Quantum Bio-Informatics III, Eds. L. Ac

cardi

et

al.

(Singapore:

World

Scientific, 2010).

10. M.

Paris,

J.

Rahecek

, eds., Quantum State Estimation, vol. 649

of

Lecture

Notes

in

Physics, (Springer, 2004).

11. A. Jamiolkowski, M. Ohya, N.

Watanabe,

T.

Matsuok

a,

in

preparation.

12. A.

S.

Holevo, Statistical Structure

of

Quantum Theory (Springer, Berlin,

2001).

13. V. Gorini

et

al.,

J.

Math. Phys.

17

, 149 (1976).

14.

G.

Lindblad,

Comm. Math. Phys.

48,

119 (1976).

15.

A.

Jamiolkowski, J. Phys. Conf. Series

213,

012002 (2010).

16. D. Shemesh, Lin. Algebra Appl.

62,

11

(1984).

17.

H. Aslaksen, A. B. Sletsj0e Lin. Algebra Appl.

430,

1 (2009).

This page intentionally left blankThis page intentionally left blank

Quantum

Bio-Informatics

IV

eds. L.

Accardi

,

W.

Freudenb

e

rg

and

M.

Ohya

World

Scientific

Publishing

Co.

(pp.

199- 208)

CONCURRENCE

AND

ITS

ESTIMATIONS

BY

ENTANGLEMENT

WITNESSES

JACEK

JURKOWSKI

Institute

of

Physics, Nicolaus Copernicus University,

Grudzigdzka

5/7,

87

- 100 Torun, Poland

The

notion

of

concurrenc

e

and

its

role

as

a

measure

of

quantum

correlations

for

both

pur

e

and

mixed

states

is

recalled.

However, for

mixed

states

concurr

ence

is

hard

to

compute

and

some

estimations

of

it

are

necessary.

It

has

been

demonstrated

how

to

use

entanglement

witnesses

in

this

procedure.

In

particular,

it

has

been

shown

that

each

entanglement

witness

detecting

given

bipartite

entangl

ed

state

provides

an

estimation

of

its

concurrence.

The

results

are

illustrated

by

several

exampl

es.

1.

Introduction

The

phenomenon

of

entanglement

between

components

of

a

quantum

sys-

tem

remains still

at

the

heart

of

many

investigations

in

quantum

informa-

tion

theory

1,2.

The

notion

of

entanglement

was

introduced

to

characterize

non-classical correlations between systems

appearing

in

quantum

theory.

Although

a

great

progress

in

quantum

information

theory

has

been

achieved

some problems concerning

detecting

and

quantifying

entanglement

are

still

open

3,4,5.

In

particular,

there

is still no general

algorithm

which enables

to

decide if a given mixed

quantum

composite

state

is

entangled

or

not.

Concerning quantifying

entanglement,

there

were

introduced

various mea-

sures

6,7,8

,9,10,

11,12

but

these which fulfill some reasonable conditions

are

hard

to

compute

for mixed

states

and

multipartite

systems. Hence,

there

is a place for

methods

which give

estimations

of

entanglement

measures

13,14,15,16,17,18.

The

most

welcome

are

procedures which for a given family

of

states

parametrized

by x allow

to

construct

estimations

as analytical

functions

of

x.

In

the

paper,

we

focus

on

some

particular

method

of

estimation

based

on

the

notion

of

an

entanglement

witness

(EW)

1,2

and

we use

it

to

estimate

a

measure

of

entanglement

called concurrence

8.

We show

that

EWs

can

be

used

not

only

to

detect

entanglement

but

to

quantify

it

as well. We

199

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